Ab Initio Study of the Friction Mechanism of Fluorographene and

May 30, 2013 - Fluorographene and graphane exhibit low friction because ... mechanism of atomic-scale friction in various graphene-based layered mater...
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Ab Initio Study of the Friction Mechanism of Fluorographene and Graphane Lin-Feng Wang, Tian-Bao Ma,* Yuan-Zhong Hu, Hui Wang, and Tian-Min Shao State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China ABSTRACT: The atomic-scale friction in fluorographene and graphane is investigated employing the density functional theory calculation including the dispersion correction (DFTD). Fluorographene and graphane exhibit low friction because of the low interlayer interaction induced by the repulsive electrostatic forces between F or H atoms at the interfaces. Moreover, owing to the less electronegativity of H atoms than F atoms, the variation of interlayer spacing and system energy corrugation, which is predominantly determined by the variation of dispersion energy during both loading and sliding processes, as elucidated in this paper, is found to be much larger in graphane, which well explains the larger friction in graphane than in fluorographene. The more charged F atom in fluorographene diminishes the variation of the interlayer spacing, which finally decreases the variation of the dispersion energy in the fluorographene. The study theoretically reveals the general mechanism of atomic-scale friction in various graphene-based layered materials where the interlayer interaction is dominated by van der Waals and electrostatic interactions, which shed light on friction reduction and the design of new lubricant materials.

1. INTRODUCTION Reducing the energy dissipation and friction is one of the main challenges to tribological researchers for decades. The progressing comprehension of atomic-scale friction provides insight into the mechanism of energy dissipation.1−4 The friction between atomically smooth surfaces is believed to depend a lot on the interfacial interactions usually described in terms of potential energy surfaces, which defines research directions for friction reduction.5−9 In most lamellar materials, the van der Waals and electrostatic interactions between adjacent layers dominant the interfacial interaction and consequently controls the friction. Taking graphite as an example, large energy dissipation and friction caused by stick− slip will take place if the graphene layers are in the commensurate state, while continuous sliding arises as one of the layers at the interface turns an angle to smooth the potential energy surface, which leads to superlubricity, a state with nearzero friction.10−12 Therefore, energy dissipation and friction reduction or even control can be achieved through exerting balance between the van der Waals and electrostatic interactions, especially for layer-structured materials. The usually used method, chemical modification, generally alters the interfacial atomic structure, the interlayer interaction, and consequently friction.13−16 Two graphene derivatives, fluorographene17 and hydrogenated graphene (graphane),18 are reported to reduce the interlayer interaction due to the atoms facing each other at the interface being likely charged.19−21 Though the two systems are believed to experience continuous sliding during the tribological process with an absence of stick− slip behaviors,2 the details of the friction properties and the corresponding mechanism still remain unknown. In this paper, © XXXX American Chemical Society

deeper investigations are conducted to uncover the friction mechanism of fluorographene and graphane and to reveal the difference between them. We hope this study increases our understanding of the atomic-scale friction in layered materials caused mainly by weak van der Waals and electrostatic type interactions and provides useful data for friction reduction and graphene-based material developments.

2. METHODOLOGY Density functional theory (DFT) is widely used to examine the interatomic interaction and to predict electronic structures. Even so, the generalized gradient approximation (GGA) and local density approximation (LDA) for the exchangecorrelation functional are unable to describe well the dispersive interaction (van der Waals interaction). In graphene-based material systems, the dispersive interaction or electrostatic interactions determine the interlayer structures and properties of the models. Thus, a correction method for long-range dispersive interaction, derived by Grimme,22 is included in our DFT calculations (DFT-D method). The Vienna Ab initio Simulation Package (VASP)22−24 is employed to calculate the atomic and electronic structures and energies of our graphenebased materials. The exchange-correlation functional is represented with the GGA in the form of the Perdew− Burke−Ernzerhof (PBE)25 functional. The projector-augmented wave (PAW)26,27 method is used to describe the Received: January 31, 2013 Revised: May 28, 2013

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interaction of valence electrons with atomic cores. The cutoff energy is 400 eV. The k points are generated through the Monkhorst−Pack (MP) algorithm.28 The integration of the Brillouin zone is performed using a 9 × 7 × 1 MP grid and tetrahedron smearing method. The energies are calculated to converge to the accuracy of 1.0 × 10−4 eV. To investigate the interlayer interactions and hence the friction properties, the computational cell is composed of two layers of fluorographene (FG) or hydrogenated graphene (graphane, HG) (shown in Figure 1). The cell with a length of 30 Å in the z direction leaves a vacuum region of more than 15 Å, which prevents interaction between the cell and its periodic image in the z direction. After the cell is fully optimized, the system energy experienced during the top layer sliding relative to the lower one is calculated. The atoms are relaxed in the z direction every time the top layer translates to a relatively different position. The potential energy surface (PES) is then constructed and derived from the system energy with the system energy E compared with the lowest energy of the system, which is similar to the methods described elsewhere.3,29,30 For the purpose of studying the load effect on the friction along the considered path, we investigate the variation of the system energy corrugation (the difference between the maximum and minimum energies) and the maximum static lateral force under different loads. The top F (H) atoms on the top layer and the bottom F (H) atoms on the lower layer are constrained, while other atoms are free when calculations are conducted under a series of given interlayer distances or different displacement along the path. Thus, as stated in refs 2 and 30, the system energy E(p,d) (p and d represent sliding path and interlayer distance, respectively) at each mesh point can be obtained; then, the lateral force component along the sliding path or friction force Ff(p,d) and the force component in the z direction or normal load Fn(p,d) can be derived. Eventually, the matrices of all data are spline interpolated. Therefore, the variation of system energy and static lateral force can be derived as one layer slides along the path under a given constant load.

Figure 1. Configuration and the corresponding potential energy surface (PES) of the (a) FG and (b) HG systems (in meV/atom) as a function of relative displacement of the two layers in the x and y directions. Arrows on the PESs show the energetically favored paths, and L and H indicate the low and high energy points of the paths, respectively. Green and gray balls represent carbon atoms; the light blue and white balls indicate fluorine and hydrogen atoms, respectively.

Figure 2. The energy corrugation of 0.31 meV/atom for the FG system is about one-third of that for the HG system, which

3. RESULTS AND DISCUSSION The optimized lengths of the C−F bond and C−C bond are 1.38 and 1.58 Å, respectively, and the interlayer spacing is about 5.76 Å. For the HG system, the length of C−H and C−C bonds and the interlayer distance are approximately 1.11, 1.53, and 4.4 Å, respectively, which agree well with other theoretical and experimental values.31,32 The PESs, each of which is mapped as a function of relative displacement of the two layers of the FG or HG system in the x and y directions, are shown in Figure 1. For convenience, the contour plot is four times the size of the computational cell with intrinsic periodicity. The maxima of the system energies marked in Figure 1 show that the FG system experiences lower energy than the HG system when the top layer is sliding relative to the lower one, which also indicates possible lower friction in the FG system. In fact, the system will generally take the energetically favored path during the sliding motion. That is, the FG or HG system will take the path shown by the arrows when the top layer slides relative to the lower layer in the x direction, as shown in Figure 1. Therefore, these two paths in FG and HG systems are chosen to study the friction of the two systems. The variation of system energy with relative displacement under the load of zero for the FG and HG systems is shown in

Figure 2. Variation of the energies of the FG and HG systems with the sliding displacement along the paths marked in Figure 1.

means that a much larger energy barrier should be overcome for the HG system to slide. The derivative of the energy with respect to displacement is the static lateral force along the sliding direction, which is expressed as f = −dE/dD, where D is the displacement. The static lateral force can be approximated to the static friction force, which is directly related to friction properties.30,33 We deem that the estimated most negative value of the static lateral force can be regarded as the maximum resistance to the sliding along the considered path. The energy corrugation and the absolute value of the most negative static lateral force for the two systems are compared in Table 1. The values indicate that the HG system experiences the largest static lateral force of 1.80 meV/Å per atom, which is significantly B

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Table 1. Comparison of System Energy Corrugation (Emax − Emin), Static Lateral Force, f m, and Shear Strength, τ, for the FG, HG, and Graphene Systems systems

energy corrugation Emax − Emin (meV/atom)

static lateral force f m (meV/Å per atom)

shear strength τ (GPa)

FG HG graphene

0.31 0.96 1.41

0.57 1.80 2.71

0.124 0.415 0.331

Because no bond breaks or forms during the sliding, it can be approximately assumed that the intralayer energy does not change. This is demonstrated as well by our calculations indeed (see the variation of the energy of the monolayer (namely, the intralayer energy) Lm and Hm shown in Figure 4), which

larger than that of the FG system. The maximum static lateral force here can be approximately regarded as the Fmax in ref 34, where the authors think that the average kinetic friction force is in proportion to Fmax, given the similar slip distance and lateral stiffness of the sliding body. This is to say that the HG system exhibits higher friction than the FG system. As depicted in Table 1, the energy corrugation, maximum static lateral force, and hence the shear strength along the energetically favored path of graphene are listed for comparison with the FG and HG systems. The shear strength along the chosen direction is obtained as τ = |f m/A|; here, f m is the maximum static lateral force and A is the area of the computational cell in the x−y plane. The graphene system has the largest energy corrugation and maximum static lateral force per atom, approximately 5 times that of the FG system. The shear strength of graphene is also much higher than that of FG, although the morphology of graphene as a single atomic layer is theoretically smoother than FG. In the FG system, fluorine atoms face each other at the sliding interface. The interlayer interaction is then reduced by the repulsion originating from the atoms with like charge. It was reported that smooth sliding rather than stick−slip behavior occurs in the FG system.2 Therefore, it is the lower energy corrugation in the FG system that results in a lower friction and energy dissipation. Figure 3 shows the variation of the energy corrugation and the maximum static lateral force with load from 1 to 2 GPa for

Figure 4. Variations of the system energies and dispersion energies with load for the (a) FG and (b) HG systems. Ls, Ld, Lm, Hs, Hd, and Hm represent the system energy, dispersion energy, and the energy of the monolayer (namely, the intralayer energy) at the point of L, and the system energy, dispersion energy, and the energy of the monolayer at the point of H, respectively. The energy values shown in the figure are comparative values; the least energy for each set (Ls/Hs, Ld/Hd, and Lm/Hm) of curves is set to be the zero energy point.

indicates high stiffness of the monolayer against loading and sliding. Therefore, the variation of the system energy can be mainly attributed to the change of interlayer interaction in these two systems, especially in the situation of quasi-static or very low velocity sliding. As mentioned above, the electrostatic and the long-range dispersive interaction between atoms of the two layers compose the interlayer interaction. Therefore, the relationship between the variation of system energy (ΔE), the electrostatic interaction energy (ΔEc), and the dispersion energy (ΔEd) can be approximately expressed as

Figure 3. Variations along the chosen paths of (a) the system energy corrugations and (b) the maximum static lateral forces with load for the FG and HG systems.

the FG and HG systems. Compared to the FG system, a nearly twice higher energy should be overcome for the HG system to slide at the different loads shown in Figure 3a. It is also demonstrated here that the higher load causes higher energy corrugation. As shown in Figure 3b, the maximum static lateral force of the HG system is much larger than that of the FG system, which also claims the lower friction in the FG system.

ΔE = ΔEc + ΔEd

(1)

The dispersion energy Ed is given by22 C

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Ed = −s6

N

∑ ∑ i=1 j=i+1

C6ij

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The electrostatic force between the two layers in the system here is repulsive force. It is understandable that the electrostatic interaction energy increases to generate a larger repulsive force to balance the load as the load increases. Figure 5 shows the

1

R ij6 1 + e−d(R ij / R r − 1)

(2)

Here, s6 is a global scaling factor, N represents the number of atoms in the system, Cij6 is the dispersion coefficient for atom pair ij, given by Cij6 = (Ci6Cj6)1/2, and Rij and Rr denote the interatomic distance and the sum of atomic van der Waals radii, respectively. d is a constant. The C6 parameters and van der Waals radii R0 for elements H, C, and F are shown in Table 2.22 Table 2. C6 Parameters (in J nm6 mol−1) and van der Waals Radii R0 (in Å) for Elements H, C, and Fa

a

element

C6

R0

H C F

0.14 1.75 0.75

1.001 1.452 1.287

Reference 22.

The electrostatic force between two atoms can be expressed as Fc = −KQiQj/R2ij, where K is a constant, and Qi and Qj represent the charges on atoms i and j, respectively. The electrostatic interaction energy Ec, the integral for the electrostatic force, is then given by N−1

Ec =

N

∑ ∑ i=1 j=i+1

KQ iQ j

1 R ij

(3)

As stated previously, determining ΔE of a system is crucial to the friction characterization. Quantitative study of the variation of the system energy and the dispersion energy with load is shown in Figure 4. The dispersion energy of the FG system increases during the sliding of the top layer from L to H under a certain constant load. The system energy also increases, though a little less than the dispersion energy (see Figure 4a); therefore, according to the eq 1, the electrostatic interaction energy is supposed to decrease, which is consistent with the predicted opposite variation trends of the dispersion energy and electrostatic interaction energy according to eqs 2 and 3, because the interlayer distance also increases from L to H, as depicted in Figure 6. A similar variation tendency is observed in the HG system, but the variation of the HG is larger than that of the FG system (shown in Figures 4b and 6). Take the load of 5 GPa for instance, as the top layer slides from L to H, the variation of the dispersion energy for the HG system, 97.71 meV, is three and a half times that of the FG system. The system energy follows to increase as well; though its variation in the HG system is smaller compared to the dispersion energy, it is 2 times larger than that of the FG system. This declares the dominating role of dispersion energy change (ΔEd) in determining the variation of system energy (ΔE). The system energy follows the dispersion energy to change when the two systems slide under constant load; that is, larger variation of the dispersion energy leads to larger variation of the system energy. Therefore, the larger variation of the system energy in the HG system due to the larger variation of the dispersion energy causes larger friction. It is demonstrated here that the HG system is more responsive during the sliding of the system. It can also be seen from Figure 4 that, at both positions of L and H, as the load increases, the dispersion energy decreases and the system energy increases, which also indicates the increase of the corresponding electrostatic interaction energy.

Figure 5. Charge differences of the C−F bond between the loads, where the charge distribution under the load of −0.76 GPa for FG and −1.12 GPa for HG system is used as the zero point for the charge difference calculation. (Left panel) Charge difference (in e/Å3) corresponding to the white box area in the inset, which shows atomic structure and charge distribution on a slice perpendicular to the y axis, under the load of −0.25 GPa for FG and −0.41 GPa for HG system. The position of F or H atom is shown, while the C atom is approximately at the bottom of the charge difference images. (Right panel) The variations of the charge differences (e/Å) along the line (marked by the white arrows) of the C−F or C−H bond are shown.

charge difference of the C−F bond (or C−H bond) between the loads shown in the right panel and the load of −0.76 GPa (or −1.12 GPa for the HG system). It is known that the difference in electronegativity between the C and F atoms is larger than that between the C and H atoms. With more charges on atoms, the interatomic distance is larger for the same electrostatic force. This explains the reason for the larger equilibrium interlayer distance for the FG system than the HG system, which is consistent with the quantitative results given at the beginning of this Results and Discussion section. The charge difference in the FG system indicates that the charge between the F atom and the C atom increases as the load increases. This manifests that the C−F bond is enhanced and hence that more charge is transferred to the F atom. Thus, the repulsive electrostatic force will further increase to bear the increasing load. It is similar in the HG system, but the charge variation between the C and H atoms is about 1 order of magnitude smaller than that between the C and F atoms, which indicates that the variation of charge on the F atoms is much larger than that on the H atoms during the loading process. As shown in Figure 6, the variation of the interlayer spacing for the D

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the density functional theory calculation including the dispersion correction (DFT-D). The FG and HG systems have smaller energy corrugation during the sliding along the energetically favored path because of the lower interlayer interaction induced by the repulsive electrostatic forces between F or H atoms at the interfaces, which then leads to lower friction. Furthermore, owing to the less electronegativity of H atoms than F atoms, larger variation of interlayer spacing and system energy corrugation during both loading and sliding processes is observed in graphane. It is further elucidated that the energy corrugation is predominantly determined by the variation of dispersion energy, not the variation of intralayer covalent bonding energy, showing that the monolayer is rather stiff against loading and sliding. The obtained larger energy corrugation in the graphane system results in larger friction than that in fluorographene. Deeper investigation shows that the smaller variation of the interlayer spacing during sliding under constant load, which decreases the variation of the dispersion energy in the fluorographene, could interpret the relatively lower friction in fluorographene than that in graphane. The analysis in this work could also explain the lower friction in the fluorated DLC than that in the hydrogenated DLC. Other than proposing fluorographene as a new superlubricious material, the study also theoretically reveals the general mechanism of atomic-scale friction in various layered materials where the interlayer interaction is dominated by van der Waals and electrostatic interactions, which shed light on friction reduction and the design of new lubricant materials.

Figure 6. Variations of interlayer distances at the positions of L and H marked in Figure 1 with load for the FG and HG systems.

FG system is significantly less than that of the HG system during the sliding under constant loads. Take the load of 5 GPa for instance, the variation of the interlayer spacing for the HG system is 0.106 Å, whereas that for the FG system is about 0.045 Å. Moreover, the interlayer distance decreases more slowly as the load increases for FG in the considered loading range, which may be a result of larger charge variation compared with the HG system. The larger interlayer distance and its smaller variation for FG cause a smaller variation of the dispersion energy. As elaborated previously, the system energy corrugation has a positive correlation with the variation of the dispersion energy. Therefore, the relatively smaller variation of dispersion energy in the FG system leads to smaller system energy corrugation and hence lower friction. Chemical modifications of surfaces or interfaces by elaborate design can tune the atomic-scale friction and system energy dissipation. In this work, both the fluorination and the hydrogenation lower the friction of graphene. Saturating the dangling bonds on the clean diamondlike carbon (DLC) film also decreases the friction in DLC.35 The friction of the fluorated and hydrogenated DLC is studied and compared in refs 36 and 37, where tribofilm is induced and transferred to the counterpart surface, resulting in F atoms (or H atom in the hydrogenated DLC) facing each other at the interface. The experimental result shows that the fluorated DLC exhibits lower friction than the hydrogenated DLC. A superlow friction state has also been achieved by fluorination of DLC films,38 which could also be explained using the theory proposed in this work. On the basis of these experimental and theoretical findings, there are reasons to also expect fluorographene as a new superlubricious material. It should be mentioned that, in real MEMS/NEMS devices, where fluorographene or graphane materials are expected to be used as a nanolubricant, the layered structure should be large enough to cover and lubricate the surface of sliding parts. Therefore, there is a necessity to learn the size-dependent mechanical and tribological properties of these graphene-based materials. Also, the instabilities or dynamic stick−slip behavior during friction, which goes beyond the scope of the present quasi-static calculations, should also be learned by combining methods of both ab initio calculations and molecular dynamics simulations, which are important directions in future studies.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +861062788310. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 51005129, 51075226, and 51021064). The authors also gratefully acknowledge the support from the Young Talents Program of China and Tsinghua National Laboratory for Information Science and Technology.

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